# Control for Schrödinger operators on 2-tori: rough potentials

### Jean Bourgain

Institute for Advanced Study, Princeton, United States### Nicolas Burq

Université Paris-Sud, Orsay, France### Maciej Zworski

University of California, Berkeley, USA

## Abstract

For the Schr\"odinger equation, $(i∂_{t}+Δ)u=0$ on a torus, an arbitrary non-empty open set $Ω$ provides control and observability of the solution: \( \| u |_{ t = 0 } \|_{ L^2 ( \T^2 ) } \leq K_T \| u \|_{ L^2 ( [0,T] \times \Omega )} \) . We show that the same result remains true for $(i∂_{t}+Δ−V)u=0$ where \( V \in L^2 ( \T^2 ) \) , and \( \T^2 \) is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for \( V \in C ( \T^2) \) and conjectured for \( V \in L^\infty ( \T^2 ) \) . The higher dimensional generalization remains open for \( V \in L^\infty ( \T^n ) \)

## Cite this article

Jean Bourgain, Nicolas Burq, Maciej Zworski, Control for Schrödinger operators on 2-tori: rough potentials. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1597–1628

DOI 10.4171/JEMS/399